Question

What is the value of `b` that would make this equation true `b\root[ 3] { 64a ^ { \frac { b } { 2} } } = ( 4\sqrt { 3} a ) ^ { 2}`?

Answer 1

`b=12`

Explanation:

There are several ways to see this. Here's one:

Given:

`b root(3)(64a^(b/2)) = (4sqrt(3)a)^2`

Cube both sides to get:

`64 b^3 a^(b/2) = (4sqrt(3)a)^6 = 4^6 * 3^3 a^6`

Equating powers of `a` we have:

`b/2 = 6`

Hence:

`b = 12`

To check, divide both ends by `4^3 = 64` to get:

`b^3 a^(b/2) = 4^3 * 3^3 a^6 = 12^3 a^6`

So looking at the coefficient of `a^(b/2) = a^6`, we have `b^3 = 12^3` and hence `b=12` works.